Mathematics 9709 · syllabus 6.1
Cambridge A Level Mathematics 9709 — The Poisson distribution topical past papers
The Poisson distribution Po(λ) for random events in time or space, its use to approximate a binomial, and its own approximation by a normal.
- Questions
- 164
- Years
- 2012–2026
- With mark scheme
- 100%
- Questions 2022–2026
- 57
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What The Poisson distribution covers
The Poisson distribution models the number of times an event happens in a fixed interval of time or space, when events occur singly, independently and at a constant average rate: calls to a help desk per hour, or flaws per metre of cable. In Paper 6 (section 6.1) you write X ~ Po(λ), use P(X = r) = e^(−λ)λʳ/r!, and use the fact that its mean and variance are both λ. Scaling is common: a rate of 3 per hour means a mean of 1 in 20 minutes.
Two approximations complete the topic. When n is large and p is small, roughly n > 50 and np < 5, B(n, p) can be approximated by Po(np). When λ is large, roughly λ > 15, Po(λ) can be approximated by N(λ, λ), with a continuity correction because a discrete count is being treated as continuous.
Paper 6 assumes the content of Paper 5, and this section leans on it directly: the Poisson approximation replaces the binomial distribution of section 5.4, and the normal approximation uses the standardisation of section 5.5, so both need to be secure first. The Poisson distribution returns in linear combinations, where the sum of independent Poisson variables is itself Poisson, and in hypothesis tests on a single Poisson observation.
In the syllabus
The headings this topic has in the Cambridge International 2026–2027 syllabus, as the board prints them.
- 6.1 The Poisson distribution
What you should be able to do
- Calculate Poisson probabilities, including 'at least' and 'at most' cases
- Scale the mean λ to a different interval of time or space
- State and explain the conditions for a Poisson model in context
- Use the equality of mean and variance to judge whether a Poisson model is plausible
- Approximate a binomial distribution by a Poisson distribution when n is large and p is small
- Approximate a Poisson distribution by a normal one, with a continuity correction
Key ideas
- X ~ Po(λ): P(X = r) = e^(−λ)λʳ/r! for r = 0, 1, 2, …
- E(X) = Var(X) = λ
- B(n, p) ≈ Po(np) when n > 50 and np < 5, approximately
- Po(λ) ≈ N(λ, λ) when λ > 15, approximately, with a continuity correction
- If X ~ Po(λ) and Y ~ Po(μ) are independent, then X + Y ~ Po(λ + μ)
How The Poisson distribution has been examined
From 2022 to 2026, The Poisson distribution was on Paper 6 · Probability & Statistics 2 in 14 of 14 exam series, worth about 28% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 6 · Probability & Statistics 2 | 57 | 28% | 14 of 14 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 9.
Filed in Momentum under: Poisson Distribution.
A worked example
Emails arrive at a help desk at random, at an average rate of 3 every 10 minutes. Find the probability that at least 2 emails arrive in a given 5-minute period, and the probability that exactly 4 arrive in a given 20-minute period.
- The rate is 3 per 10 minutes, so in 5 minutes X ~ Po(1.5).
- P(X ≥ 2) = 1 − P(X = 0) − P(X = 1) = 1 − e^(−1.5)(1 + 1.5).
- = 1 − 2.5 × 0.22313 = 1 − 0.55783 = 0.44217.
- In 20 minutes the mean is 6, so Y ~ Po(6) and P(Y = 4) = e^(−6) × 6⁴/4!.
- = 0.0024788 × 1296/24 = 0.0024788 × 54 = 0.13385.
Answer: P(at least 2 in 5 minutes) = 0.442; P(exactly 4 in 20 minutes) = 0.134
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Not rescaling λ for a new interval: a rate of 4 per hour means λ = 2 for 30 minutes.
- Stating the conditions for a Poisson model without context: 'faults occur independently and at a constant average rate along the cable' is what earns the mark.
- Leaving out P(X = 0) when finding P(X ≤ r) or P(X ≥ 1): the distribution starts at r = 0.
- Using N(λ, λ) with standard deviation λ: the standard deviation is √λ.
Exam technique
- Write the distribution with its parameter for the interval in the question, such as X ~ Po(2.5), before calculating.
- For 'at least' probabilities, use 1 − P(X ≤ r − 1) and show each term you subtract.
- When you use an approximation, state the condition it needs and show that the numbers given meet it.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.